Subgroups — C₂×#20244329 #26220725

Order 12032.

1438 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic
3	#20244329	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688878	C₂	1	4	yes	maximal
5	C₂²×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	C₂×#13093935	—	2	4	no	—
11	C₂×#13093935	C₂²	1	4	yes	—
12	#15688878	C₂²	1	16	yes	—
13	C₂³×Q₈×C₄₇	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₄₇	—	2	4	no	—
15	C₂×C₄×Q₈×C₄₇	—	2	4	no	—
16	C₂×C₄×Q₈×C₄₇	C₄	1	4	yes	—
17	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	—
18	C₂²×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄₇⋊C₈	—	2	4	no	—
23	C₂²×C₄₇⋊C₈	—	2	4	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	—	2	8	no	—
26	C₂²×C₄₇⋊C₈	Q₈	1	8	yes	complemented
27	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
29	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
30	#13093935	—	2	8	no	—
31	C₂²×Q₈×C₄₇	—	2	4	no	—
32	C₂²×Q₈×C₄₇	—	2	8	no	—
33	C₂²×Q₈×C₄₇	C₂×C₄	1	4	yes	—
34	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₄×Q₈×C₄₇	—	2	16	no	—
40	C₄×Q₈×C₄₇	—	2	8	no	—
41	C₄×Q₈×C₄₇	—	2	8	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
46	C₂×C₄²×C₄₇	—	2	4	no	—
47	C₂×C₄²×C₄₇	—	2	4	no	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₇	D₄	1	4	yes	—
51	C₂×C₄²×C₄₇	D₄	1	4	yes	—
52	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
53	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
54	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×Q₈×C₄₇	—	2	32	no	—
56	C₂×Q₈×C₄₇	—	2	8	no	—
57	C₂×Q₈×C₄₇	—	2	16	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	2	8	no	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
66	C₂²×C₄×C₄₇	M₁₆	1	8	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
75	C₂²×C₄×C₄₇	#41	1	4	yes	—
76	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
77	C₄⋊C₄×C₄₇	—	2	16	no	—
78	C₄⋊C₄×C₄₇	—	2	8	no	—
79	C₄²×C₄₇	—	2	8	no	—
80	C₄²×C₄₇	—	2	4	no	—
81	C₄²×C₄₇	—	2	4	no	—
82	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
83	C₄²×C₄₇	C₂×D₄	1	4	yes	—
84	C₄²×C₄₇	C₂×D₄	1	4	yes	—
85	C₂×C₄₇⋊C₈	—	2	16	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	2	8	no	—
90	C₂×C₄₇⋊C₈	—	4	8	no	—
91	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	complemented
92	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
94	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
95	C₂³×C₄₇	#131	1	4	yes	—
96	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
97	Q₈×C₄₇	—	2	32	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	8	no	—
103	C₂×C₄×C₄₇	—	2	16	no	—
104	C₂×C₄×C₄₇	—	2	8	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	8	no	—
107	C₂×C₄×C₄₇	—	4	8	no	—
108	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
110	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
112	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
113	C₂×C₄×C₄₇	#124	1	4	yes	—
114	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
115	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
116	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
120	C₄₇⋊C₈	—	2	8	no	—
121	C₄₇⋊C₈	—	4	4	no	—
122	C₂×#1985	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	#391	1	4	yes	—
124	C₂²×C₄₇	#407	1	4	yes	—
125	C₂²×C₄₇	#408	1	4	yes	—
126	C₂²×C₄₇	#446	1	16	yes	—
127	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₄₇	C₂×C₄×Q₈	1	2	yes	—
129	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
130	C₂²×C₄₇	C₂×#131	1	2	yes	—
131	C₄×C₄₇	—	2	4	no	—
132	C₄×C₄₇	—	2	8	no	—
133	C₄×C₄₇	—	2	4	no	—
134	C₄×C₄₇	—	4	4	no	—
135	C₄×C₄₇	#391	1	4	yes	—
136	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
137	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
138	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
139	C₂²×C₄×Q₈	D₄₇	1	1	yes	characteristic
140	C₂²×C₄⋊C₈	—	47	1	no	—
141	C₂²×#97	—	47	1	no	—
142	#1985	—	47	8	no	—
143	C₂×#336	—	47	4	no	—
144	C₂×C₄₇	#1985	1	8	yes	complemented
145	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
146	C₂×C₄₇	C₂×#407	1	1	yes	characteristic
147	C₂×C₄₇	C₂×#408	1	1	yes	characteristic
148	C₂×C₄₇	C₂×#446	1	4	yes	—
149	C₂³×Q₈	Dic₄₇	1	1	yes	characteristic
150	C₂³×C₈	—	47	2	no	—
151	C₂³×C₈	—	47	2	no	—
152	C₂×C₄×Q₈	—	2	4	no	—
153	C₂×C₄×Q₈	—	2	4	no	—
154	C₂×C₄×Q₈	Dic₄₇	1	4	yes	—
155	C₂×C₄×Q₈	C₂×D₄₇	1	4	yes	—
156	C₂²×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
157	C₂²×C₄⋊C₄	Dic₄₇	1	2	yes	—
158	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
159	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
160	C₂×C₄⋊C₈	—	47	4	no	—
161	C₂×C₄⋊C₈	—	94	4	no	—
162	C₂×#97	—	47	4	no	—
163	C₂×#97	—	94	4	no	—
164	#336	—	47	16	no	—
165	C₄₇	C₂×#1985	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₈	—	2	8	no	—
167	C₂²×Q₈	—	2	4	no	—
168	C₂²×Q₈	C₂×Dic₄₇	1	4	yes	—
169	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
170	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
171	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
172	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
173	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
174	C₂²×C₈	—	47	8	no	—
175	C₂²×C₈	—	47	4	no	—
176	C₂²×C₈	—	47	4	no	—
177	C₂²×C₈	—	94	4	no	—
178	C₂²×C₈	—	94	4	no	—
179	C₂²×C₈	—	94	8	no	—
180	C₂²×C₈	—	94	4	no	—
181	C₄×Q₈	—	2	16	no	—
182	C₄×Q₈	—	2	8	no	—
183	C₄×Q₈	—	2	8	no	—
184	C₂×C₄⋊C₄	—	2	8	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
187	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
188	C₂×C₄²	—	2	4	no	—
189	C₂×C₄²	—	2	4	no	—
190	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
191	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
192	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
193	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
194	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
195	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
196	C₄⋊C₈	—	94	8	no	—
197	#97	—	94	8	no	—
198	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
199	C₂×Q₈	—	2	32	no	—
200	C₂×Q₈	—	2	8	no	—
201	C₂×Q₈	—	2	16	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	#10625329	1	8	yes	—
209	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
210	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
211	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
212	C₂²×C₄	#10625354	1	4	yes	—
213	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
214	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
215	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
216	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
217	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
219	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
220	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
221	C₂×C₈	—	47	8	no	—
222	C₂×C₈	—	94	16	no	—
223	C₂×C₈	—	94	4	no	—
224	C₂×C₈	—	94	4	no	—
225	C₂×C₈	—	94	4	no	—
226	C₂×C₈	—	94	8	no	—
227	C₂×C₈	—	188	8	no	—
228	C₄⋊C₄	—	2	16	no	—
229	C₄⋊C₄	—	2	8	no	—
230	C₄²	—	2	8	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	4	no	—
233	C₄²	C₂²⋊Dic₄₇	1	8	yes	—
234	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
235	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
236	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
237	C₂³	Q₈×Dic₄₇	1	4	yes	—
238	C₂³	C₂×#10625329	1	4	yes	—
239	C₂³	#13094031	1	4	yes	—
240	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
241	Q₈	—	2	32	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	16	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
253	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
254	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
255	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
256	C₂×C₄	#13094000	1	8	yes	—
257	C₂×C₄	#13094013	1	4	yes	—
258	C₂×C₄	#13094022	1	4	yes	—
259	C₂×C₄	C₂×#10625329	1	8	yes	—
260	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
262	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
263	C₂×C₄	C₂×#10625354	1	2	yes	—
264	C₈	—	94	8	no	—
265	C₈	—	188	4	no	—
266	C₂²	#15689030	1	4	yes	—
267	C₂²	#15689087	1	4	yes	—
268	C₂²	#15689090	1	4	yes	—
269	C₂²	#15689254	1	16	yes	—
270	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
271	C₂²	C₂×Q₈×Dic₄₇	1	2	yes	—
272	C₂²	C₂²×#10625329	1	2	yes	—
273	C₂²	C₂×#13094031	1	2	yes	—
274	C₄	—	2	4	no	—
275	C₄	—	2	8	no	—
276	C₄	—	2	4	no	—
277	C₄	—	4	4	no	—
278	C₄	#15689030	1	4	yes	—
279	C₄	C₂×#13094000	1	2	yes	—
280	C₄	C₂×#13094013	1	1	yes	characteristic
281	C₄	C₂×#13094022	1	1	yes	characteristic
282	C₂	#20244329	1	8	yes	direct product factor
283	C₂	C₂×#15689030	1	1	yes	characteristic
284	C₂	C₂×#15689087	1	1	yes	characteristic
285	C₂	C₂×#15689090	1	1	yes	characteristic
286	C₂	C₂×#15689254	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄×Q₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₄₇
hypocenter	C₄₇	C₂×#1985
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #20244329	C₂	direct (N × H)
26. C₂²×C₄₇⋊C₈	Q₈	semidirect
91. C₂×C₄₇⋊C₈	C₂×Q₈	semidirect
144. C₂×C₄₇	#1985	semidirect
165. C₄₇	C₂×#1985	semidirect
282. C₂	#20244329	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄⋊(C₄₇⋊C₈)	2
C₂²×#13093935	2
#20244329	2
C₂×#15688878	2
C₂²×C₄×Q₈×C₄₇	2
C₂×#1985	47

Subgroups — C2×#20244329 #26220725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244329 #26220725